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This is an extremely subjective area - there is much that can be learned but there are so many overlapping opinions and shades of interpretation that skimming through a very wide range of material and then reading in depth those that seem to be of value to you is liable to be more useful than a few explanations here, no matter how good. The internet has a vast number of commentaries and guides available on this subject and I consider that the best approach is probably steeping oneself in the range of available material and seeing what you learn that seems useful in the process.

I am personally cautious to skeptical of most composition rules, and especially so of 'the triangle rule', which turns out to be several 'rules', or none. One variant has the golden triangle as half a golden rectangle and gors from there. Others see clusters of triangles - or say you should be able to. Some can probably explain any "good" photo by adding triangles tha may or may not have any effect on the viewers perception. There are definitely aspects such as leading the eye, clustering objects or subject material in a defined shape etc which will be useful on occasion. How much so is best determine (I think) by lots of skimming through material on the subject and then trying it yourself.

Photography related triangles material - see at end

Triangles in culture:

Getting away from pure photography, but ...

This horoscope oriented site gives a range of examples of diamond and triangular structures in cultrual images and themes from various times and thnicities.

Symbols and their meanings - useful.
Note that the writer of this page has a strong Christian perspective which flavours some of his comments but still conveys well the underlying meaning and uses.
Summarised from there:

Masonic all seeing eye - see US $1 bill.

Several Alchemists symbols include a triangle

Anarchist symbol

Blair Witch - symbolic microcosmic man - Alchemists symbol

Masonic compass - maybe

Hexagram - two intersected triangles - Star of David.
(Interest: Star of Solomon is 5 pointed and looks similar at first glance.)

Dphoto are often enough an OK source of information. But I think this photographer in Photography composition tip: Enforce order through triangles - sees too many triangles in too many locations. Some of these photos would be impossible to compose in real life situations (eg ballet dancers) and others (eg elephant * man or '3 net holders + tree') could have triangles drawn in other equally valid places.

Here are 4 rules with the "diagonal lines" comment being useful and just possibly as good advice as most on this subject, and they add nice perspective with a quote by Edward Weston at the end -

“To consult the rules of composition before making a picture is a little like consulting the law of gravitation before going for a walk. Such rules and laws are deduced from the accomplished fact; they are the products of reflection.”

[Paul Cézanne](Paul Cézanne successfully used triangles in his compositions of still lifes.) successfully used triangles in his compositions of still lifes.

They may be right :-).
Les Grandes Baigneuses, 1898–1905

Paul died in 1906, even before Mickey Mouse was born. Some argue that faithful 2-D images of PD art may not themselves be PD. Wikipedia disagree, violently. Comments on this issue here

I find resources like this of immense value - a picture being worth a thousand word, you get about 250,000 words to skim though in seconds. I think that targeted image search is one of the great unsung tools available.

Are you referring to the Exposure Triangle? photo.stackexchange.com/questions/6598/… Because @mattdm answered that himself and got a bajillion upvotes. :P He may not be referring to that in this particular question.
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BBkingOct 29 '12 at 3:42

@BBking - Very possibly Matt has answers in mind. As noted above, I am not referring to one thing as people refer to many things under this heading - as I intended the few references to show. You can find many references on this subject - and the more one reads the wider the subject material becomes. I liked Edward Weston's comment - it happened to match what I'd said above, but I had not read his quote when I made my comment :-).
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Russell McMahonOct 29 '12 at 4:11

@BBking - I just looked at Matt's prior answer - No - nothing whatsoever to do with exposure triangle
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Russell McMahonOct 29 '12 at 7:57

I had a comment here which I deleted because you extended your answer. He goes on further to explain an "exposure cuboid" but the question was about the "exposure triangle". I didn't quite understand the question at first as I thought it was just a "why things in threes" in general type of question. Thanks for your answer.
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BBkingOct 29 '12 at 9:18